Maths Assertion (A) & Reason (B) MCQ

3. Assertion is correct statement but Reason is wrong statement.

Solution:

$\frac{1}{\text{y}^3}\frac{\text{dy}}{\text{dx}}+\frac{\text{x}}{\text{y}^2}=\text{x}^3$

Put $\frac{1}{\text{y}^2}=\text{z}\Rightarrow\frac{2}{\text{y}^3}\text{dy}=\text{dz}$

$\therefore\frac{\text{dz}}{\text{dx}}-2\text{xz}=-2\text{x}^3,$

which is a linear differential equation with $\text{I.F}=\text{e}^{\text{x}^2}$

$\therefore$ $\text{ze}^{-\text{x}^2}=-\int\text{e}^{\text{x}^2}2\text{x}^3\text{dx} $

$\Rightarrow\text{ze}^{-\text{x}^2}=(\text{x}^2+1)\text{e}^{\text{-x}^2}+\text{C}\Rightarrow\text{z}=\text{x}^2+1+\text{C}\text{e}^{\text{x}^2}$

$\therefore\frac{1}{\text{y}^2}=\text{x}^2+1+\text{C}\text{e}^{\text{x}^2}$

$\because\text{y}(0)=1\Rightarrow\text{c}=0$

$\therefore\text{y}^2=\frac{1}{\text{x}^2+1}\Rightarrow\text{y}=\frac{1}{\sqrt{\text{x}^2+1}}\Rightarrow\text{y}(1)=\frac{1}{\sqrt{2}}$

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

Let $=(\text{c}_1+\text{c}_2+\text{c}_3\text{e}^\text{c}4)=\text{A}\text{(Constant)}$

Then, $\text{y} = \text{Ax}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{A}\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$

$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$

1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

$\frac{\text{dy}}{\text{dx}}+\bigg(\frac{1}{\sin\text{x}}+\cot\text{x}+\frac{1}{2}\bigg)^\text{y}=\frac{1}{\text{x}}$

$\text{I.F}=\text{exp}\int\bigg(\frac{1}{\sin\text{x}}+\cot\text{x}+\frac{1}{\text{x}}\bigg)\text{dx}$

$=\text{exp In}\bigg(\text{x}\tan\frac{\text{x}}{2}\sin\text{x}\bigg)$

$=\text{x}\tan\frac{\text{x}}{2}\times2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}=\text{x}(1-\cos\text{x})$

$\therefore$ Solution is, $\text{yx}(1-\cos\text{x})=\int\frac{1}{\text{x}}\text{x}(1-\cos\text{x})\text{dx}$

$=\text{x}-\sin\text{x+c}$

2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Solution:

Let x² + y² + 2gx + 2fy + c = 0

Here, in this equation, there are three constants.

$\therefore$ Order = 3

Reason is also correct.

4. Assertion is wrong statement but Reason is correct statement.

Solution:

$\because \text{y}=(\text{c}_1\text{e}^{\text{c}2}+\text{c}_3\text{e}^{\text{c}4})\text{e}^\text{x}=\text{ce}^\text{x}$

$\therefore\frac{\text{dy}}{\text{dx}}=\text{ce}^\text{x}\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}$ (Using-(i))

$\therefore$ Order is 1.

2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Solution:

$\because\text{ y}=\text{a}\sin\text{ x}+\text{b }\cos \text{x}$

$\therefore\text{ y}=\text{a}\cos\text{ x}-\text{b }\sin \text{x}$

$\Rightarrow\text{y}"=-\text{a}\sin\text{x}-\text{b}\cos\text{x}=-\text{y}$

$\Rightarrow\text{y}''+\text{y}=0$